Final answer:
Orthogonal 3x3 matrices with all off-diagonal entries equal to 0 must have diagonal entries that are either 1 or -1, resulting in 8 possible matrices with various sign combinations.
Step-by-step explanation:
The orthogonal 3×3 matrices for which all off-diagonal entries are 0 can be obtained by considering the definition of an orthogonal matrix. An orthogonal matrix O must satisfy the equation OTO = I, where OT is the transpose of O, and I is the identity matrix. With only diagonal entries being nonzero, this product results in the squares of the diagonal elements being summed on the diagonal of the resulting matrix. Therefore, the diagonal entries of O must be either 1 or -1 to ensure that the sum is 1, which is the definition of the identity matrix.
We can conclude that the resulting orthogonal matrices with all off-diagonal entries being 0 have only ±1 on their diagonal. This gives us a total of 23 = 8 possible matrices, as we have three independent choices for the signs of the diagonal entries. Examples of such matrices include the identity matrix and a matrix with diagonal entries -1, 1, 1, which reflect along the first coordinate axis.